A019283 -id:A019283 - cp's OEIS frontend

A326181 Numbers n for which sigma(sigma(n)) = 3*sigma(n).

54, 56, 87, 95, 276, 308, 429, 446, 455, 501, 581, 611, 158928, 194928, 195072, 199950, 226352, 234608, 236432, 248325, 255678, 263504, 266192, 273050, 275415, 304575, 336903, 341162, 353675, 366575, 369425, 369843, 380463, 386313, 389463, 406565, 411725, 415925, 422303, 447587, 468743, 497333, 500993, 511829, 515267, 519557, 519677

Offset: 1

Antti Karttunen, Jun 16 2019

nonn

Any odd perfect numbers must occur in this sequence, as such numbers must be in the intersection of A000396 and A326051, that is, satisfy both sigma(n) = 2n and sigma(2n) = 6n = 3*2n, thus in combination they must satisfy sigma(sigma(n)) = 3*sigma(n). Note that odd perfect numbers should occur also in A019283.

If, as conjectured, A005820 has 6 terms, then this sequence is finite and has 756 terms. - Giovanni Resta, Jun 17 2019

Giovanni Resta, Table of n, a(n) for n = 1..756 (first 169 terms from Antti Karttunen)
Index entries for sequences where any odd perfect numbers must occur

Cf. A000203, A000396, A005820, A019283, A051027, A087943, A272027 (3*sigma(n)), A326051.

Subsequence of A066961.

isA326181(n) = { my(s=sigma(n)); (sigma(s)==3*s); };

A332457 Numbers k such that sigma(k) == 2 modulo 8 and sigma(sigma(k)) == 6 modulo 8.

Original entry on oeis.org

193, 202, 673, 1153, 1201, 1354, 1601, 1642, 1873, 2017, 2088, 2593, 2682, 2753, 3049, 3112, 3217, 3313, 3328, 3754, 3898, 4041, 4084, 4177, 4273, 4337, 4426, 4561, 5193, 5233, 5386, 5449, 5482, 5849, 6337, 6353, 6826, 6922, 7002, 7057, 7114, 7393, 7402, 7537, 7793, 8081, 8104, 8353, 8564, 8698, 8872, 9049, 9377, 9601

Offset: 1

Antti Karttunen, Feb 15 2020

nonn

That the first part of the condition is necessary for odd perfect numbers, see A332228, that the second part of the condition is necessary, see A019283 and A326181.

Cf. A000203, A019283, A051027, A326181.
Intersection of A332226 and A332456.
Cf. A332458 (a subsequence of non-primepower odd terms).

[k:k in [1..9700]| DivisorSigma(1,k) mod 8 eq 2 and DivisorSigma(1, DivisorSigma(1,k)) mod 8 eq 6]; // Marius A. Burtea, Feb 15 2020

Select[Range[10000],With[{c=DivisorSigma[1,#]},Mod[c,8]==2&&Mod[DivisorSigma[1,c],8]==6&]]  (* Harvey P. Dale, Nov 23 2024 *)

isA332457(n) = { my(s=sigma(n)); ((2==(s%8)) && (6==(sigma(s)%8))); };

A205597 Odd terms of A019278: odd n such that sigma(sigma(n))/n is an integer.

Original entry on oeis.org

1, 15, 21, 1023, 29127, 550095, 355744082763

Offset: 1

Jud McCranie, Feb 08 2012

a(8) > 4*10^12, if it exists. - Giovanni Resta, Feb 26 2020

First five terms are squarefree. Sigma(sigma(n))/n ratios for these seven known terms are: 1, 4, 3, 4, 4, 6, 4. - Antti Karttunen, Mar 19 2021

			15 is odd, sigma(15) = 24, sigma(24) = 60, and 60/15 is an integer.

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Cf. A000203, A019278, A019283, A228058, A323653.

Select[Range[1, 10^6, 2], Mod[Nest[DivisorSigma[1, #] &, #, 2], #] == 0 &] (* Michael De Vlieger, Mar 19 2021 *)

isok(n) = (n%2) && (denominator(sigma(sigma(n))/n) == 1); \\ Michel Marcus, Sep 27 2017

A247111 Integers k such that sigma(sigma(k) - k) = 2*k, where sigma is the sum of divisors, A000203.

Original entry on oeis.org

6, 28, 36, 496, 8128, 33550336, 8589869056

Offset: 1

Michel Marcus, Nov 19 2014

That is, integers k such that A072869(k) = 2*k.
All perfect numbers (A000396) belong to this sequence.
Is there another term like 36 that is not perfect?
a(8) > 10^11. - Hiroaki Yamanouchi, Sep 11 2015
a(8) <= 137438691328. - David A. Corneth, Jun 04 2021

			For k=36, sigma(sigma(36)-36) = sigma(91-36) = sigma(55) = 72, hence 36 is in the sequence.

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Cf. A000203 (sigma(n)), A000396 (perfect numbers), A001065 (sigma(n)-n), A072869 (sigma(sigma(n)-n)).

Cf. also A019283, A326181, A342922.

Select[Range[1,10000],DivisorSigma[1,DivisorSigma[1,#]-#]==2*#&] (* Julien Kluge, Sep 20 2016 *)

PARI
```
isok(n) = (sigma(sigma(n) - n) == 2*n);
```

a(7) from Michel Marcus, Nov 22 2014

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A326181 Numbers n for which sigma(sigma(n)) = 3*sigma(n).

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A332457 Numbers k such that sigma(k) == 2 modulo 8 and sigma(sigma(k)) == 6 modulo 8.

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A205597 Odd terms of A019278: odd n such that sigma(sigma(n))/n is an integer.

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A247111 Integers k such that sigma(sigma(k) - k) = 2*k, where sigma is the sum of divisors, A000203.

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